Codes of central Sturmian words

• Authors: Arturo Carpi;Aldo de Luca
• Venue: Theoretical Computer Science - The art of theory
• Year: 2005

A central Sturmian word, or simply central word, is a word having two coprime periods p and q and length equal to p + q - 2. We consider sets of central words which are codes. Some general properties of central codes are shown. In particular, we prove that a non-trivial maximal central code is infinite. Moreover, it is not maximal as a code. A central code is called prefix central code if it is a prefix code. We prove that a central code is a prefix (resp., maximal prefix) central code if and only if the set of its 'generating words' is a prefix (resp., maximal prefix) code. A suitable arithmetization of the theory is obtained by considering the bijection θ, called ratio of periods, from the set of all central words to the set of all positive irreducible fractions defined as: θ(ε) = 1/1 and θ(w) = p/q (resp., θ(w) = q/p) if w begins with the letter a (resp., letter b), p is the minimal period of w, and q = |w| - p + 2. We prove that a central code X is prefix (resp., maximal prefix) if and only if θ(X) is an independent (resp., independent and full) set of fractions. Finally, two interesting classes of prefix central codes are considered. One is the class of Farey codes which are naturally associated with the Farey series; we prove that Farey codes are maximal prefix central codes. The other is given by uniform central codes. A noteworthy property related to the number of occurrences of the letter a in the words of a maximal uniform central code is proved.